Construction of Parallel Lines

We are basically going to learn how to construct parallel lines and the steps to construct parallel lines.

What is A Parallel Line?

Parallel lines are lines which do not have a common meeting point in the same plane, however far they are extended .The following figure represents parallel lines. Sometimes you may be presented with one line and you need to create another line parallel to it through any given point. You might think to simply take a straight edge and draw a line that might seem right; however, you could not be sure that the line you constructed is actually parallel. With the use of geometry and a compass, you can easily plot additional points that will ensure the line you construct is truly parallel.

Properties of Parallel Lines

1) The corresponding angles formed by parallel lines are equal.

2) The vertically opposite angles formed by parallel lines are equal.

3) The alternate interior angles formed by parallel lines are equal.

4) The alternate exterior angles formed by parallel lines are equal.

5) The pair of interior angles on the same side of the transversal are supplementary, that is they equal to 180 degrees.

Construction of Parallel Lines:

Given: We have been a point P .

Construct : We have to construct parallel lines.

The Steps for Constructing Parallel Lines Is Quite Simple! Here Are The Steps To Construct Parallel Lines-
Step 1) Use your straightedge, and draw a transversal through the given point . This is simply a straight line which passes through the given point P and intersects with the given line. Drawing the line slanted will make the construction easier than to try to make the line in a vertical manner. Be sure that you draw the line properly above point P.
Step 2) Using the construction you can copy an angle, construct a copy of the angle formed by the transversal and the given line such that the copy will be located UP at point P. The vertex of the copied angle will be located at the point P.
Step 3) When you draw the line to complete the angle copy, you will be able to draw a line parallel to the given line.

Image will be uploaded soon

Steps To Construct Parallel Lines

Here’s an Alternate Method for constructing parallel lines:

The construction done above represents the creation of congruent corresponding angles by the lines which make them parallel. As shown in the right side, you can also copy the angle below P and to its left, which will create an alternate interior angle and also lead to the construction of parallel lines.

The corresponding angle approach is often preferred because it prevents the construction lines for the angles from bumping into another line.


Image will be uploaded soon


How To Construct Parallel Lines?

Let’s understand Step by Step to Construct Parallel Lines-

If we have to construct a line parallel to the other line from an external point all we require is a ruler and a compass and the following steps need to be kept in mind!

Given: A line segment named AB and a given point P that lies out of the line segment AB.

To construct a line that is parallel to line AB that passes through the given point P.

Step 1: You have to choose any point X on the given line segment AB and join it to point P as shown below in the diagram.

Image will be uploaded soon

 

Step 2: Considering X as the center and any suitable radius you need to draw an arc cutting the line segment PX at the point M and AB at point N respectively.

Image will be uploaded soon


Step 3: As you have P as the center and radius remains same as used in the previous step 2 you need to draw an arc EF cutting the line segment PX at point Q as shown in the figure below.


Image will be uploaded soon

Step 4: With Q as the center and same radius as we have used in Step 1, draw an arc cutting the arc EF at R as shown below in the given diagram.

Image will be uploaded soon

Step 5: Join the points R and P to draw a line segment CD as shown in the figure given below.

Image will be uploaded soon

The line segment CD is the line that needs to be parallel to the line segment AB and passes through the point P.

Note: Now you might think how to know whether a line is parallel to the other given line or not. To check whether or not two lines are parallel, we must compare their slopes which are denoted by m in the equation (y = mx+c). Two lines are parallel if and only if their slopes which are m are equal in measure.

FAQ (Frequently Asked Questions)

1. How to construct  parallel lines and do parallel lines have the same slope?

Two lines are known to be parallel if they have the same slope that is (m). Slope is denoted by the letter m. Here’s an example: suppose we need to find the slope of the line parallel to the line 4x – 5y = 12. Now to find the slope of this line we need to get the line into slope-intercept form (y = mx + c), which means we need to solve for y: The slope of the line 4x – 5y = 12 is = 4/5.

 

Parallel lines will never intersect and have the same slope. Assuming that the lines lie on the same plane parallel lines continue, literally, forever without touching. On the other hand, the slope of perpendicular lines are the negative reciprocals of each other and a pair of perpendicular lines intersects at 90 degrees.

2. What is slope intercept form and what is a congruent angle?

The slope-intercept form of the equation of a line is y = mx + c, where m is equal to the slope of the line, and c is the y-intercept. Since we are given two points, we can calculate the slope m as follows, Even if we interchange the points the slope remains the same.

 

Congruent Angles have the same angle (in degrees or radians). 

 

These angles are congruent. Such angles don't have to point in the same direction. The congruent angles don't have to be on similar sized lines. They are always equal in measure.